SPP-2410: Hyperbolic Balance Laws in Fluid Mechanics: Complexity, Scales, Randomness

Nonlinear hyperbolic balance laws are ubiquitous in the modelling of fluidmechanical processes. They enable the development of powerful numerical simulation methods that back decision-making for critical applications such as in-silico aircraft design or climate change research. However, fundamental questions about distinctive
hyperbolic features remain open including the multi-scale interference of shock and shear waves, or the interplay of hyperbolic transport and random environments. The largely unsolved well-posedness problem for multi-dimensional inviscid flow equations is deeply connected to the laws of turbulent fluid motion in the high Reynolds-number limit. Further progress requires a concerted effort of both fluid mechanics and the mathematical fields of analysis, numerics, and stochastics.

The Priority Programme is devoted to the development of new mathematical models and methods to understand  the dynamic creation of small scales and mechanisms which are either enhanced or depleted by the hyperbolic nonlinearity. It strives at a novel numerical paradigm for hyperbolic transport that can provide firm grounds for the upcoming theory of small-scale turbulence in the large Reynolds number limit. The Priority Programme will mostly evolve around three major research directions:

Novel solution concepts: This includes the analysis for hyperbolic systems arising in fluid mechanics (via e.g. generalized entropy methods, dissipative limits or probabilistic and moment-based solutions), the design of high-resolution numerics for these solution concepts, and  exploring the connections to  modern statistical turbulence modelling and perturbation/filtering techniques.

Multi-scale models and asymptotic regimes: Research includes the development and analysis of model hierarchies (e.g. Boltzmann-Euler or in statistical turbulence) and their closures that account for asymptotic flow regimes (e.g. Mach number limits). Entropy- and structure-preserving numerical methods need to be designed that allow well-balancing and preservation of asymptotic states while traversing through hierarchies and regimes by accuracy-controlled model selection.

Probabilistic models: This area comprises the analysis, numerics and uncertainty quantification for stochastic models of hyperbolic systems arising in fluid mechanics. It includes probabilistic modelling concepts to explore statistical turbulence using e.g. stochastic variational principles and the exploration of stochastic/data-driven tools for hybrid perturbation/filtering techniques. Methods of uncertainty quantification should account for preservation of hyperbolic features.